Algebra Prelim

June 2006

1. Let g, h ∈ G where G is a group. Suppose that gh = hg and that |g| = 10 and that |h| = 12. Prove that |gh| ≥ 30.

2. Prove that the symmetric group S_{5} is generated by the two permutations (12) ∘ (345) and (12345).

3. Let V be a vector space of dimension 8 over some field k. Let T : V → V be a linear transformations such that dim(ker(T)) = 1. Then:

(a) Determine the possibilities for the dimensions of T (V), T^{2}(V) = T (T (V)), T^{3}(V) = T (T^{2}(V)), . . ..

(b) Argue that if T^{n} = 0 for some n ≥ 1 then n ≥ 8 and in fact that in this case T^{8} = 0.

4. (a) Let G be a finite group. Suppose that G has exactly 5 Sylow p-subgroups for some prime p. Explain why G has an element of order 5 and an element of order 2.

(b) Now let G be a finite group such that G has exactly 1 + 2^{n} Sylow p-subgroups for some prime p and some n ≥ 1. Explain why G has an element of order q for each prime q that divides 1 + 2^{n} and also has an element of order 2.

5. Let ℜ be the field of real numbers. Consider the ring homomorphism ϕ : ℜ[x] → M_{2}(ℜ) (where M_{2}(ℜ) is the ring of 2 × 2 matrices over ℜ)

that maps r ∈ ℜ to ( \begin{matrix}r & 0 \\ 0 & r \end{matrix} ) and that maps x to ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ). Then:

(a) Find the kernel of ϕ.

(b) Find the dimension of ℜ[x]/ker(ϕ) as a vector space over ℜ

(c) Is ℜ[x]/ker(ϕ) an integral domain or not? Explain your answer.

6. Let R be a commutative ring. Suppose that for some a, b ∈ R we have as + bt = 1 for some s, t ∈ R. Then:

(a) Prove that the function r ↦ (r + (a), r + (b)) from R to R/(a) × R/(b) is surjective and that its kernel is (a) ∩ (b)

(b) Show that (a) ∩ (b) = (ab)

(c) Consider 17, 33 ∈ ZZ. Find all integers n such that n has a remainder of 3 when divided by 13 and a remainder of 5 when divided by 33.

7. Let C(x) be the field of fractions of C[x] where C is the field of complex numbers. Let ζ_{6} ∈ C be a primitive 6-th root of unity. Consider the unique homomorphism σ : C(x) → C(x) over C such that σ(x) = ζ_{6}x. Then:

(a) Find the order of σ as an element of the group of automorphism of C(x), i.e. of Aut(C(x)).

(b) Argue that C(x^{6}) is in the fixed field of σ.

(c) Show that C(x^{6}) ⊂ C(x) is a Galois extension and give its Galois group.

(d) Note that C(x^{6}) ⊂ C(x^{3}) ⊂ C(x). Using this, determine Gal(C(x)/C(x^{3})) as a subgroup of Gal(C(x)/C(x^{6})).

8. Let k be a finite field. Let f(x) ∈ k[x] be such that f(0) ≠ 0 and such that gcd(f(x), f′(x)) = 1. Prove that for some n ≥ 1, f(x) divides x^{n} − 1 in k[x].